Download App

Trigonometric Identities — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsTrigonometric Identities5 Marks
Question
If $\text{x}=\text{a}\sec\theta+\text{b}\tan\theta$ and $\text{y}=\text{a}\tan\theta-\text{b}\sec\theta,$ prove that $\big(\text{x}^2+\text{y}^2\big)=\big(\text{a}^2-\text{b}^2\big).$
✓

Answer

We have $\text{x}^2+\text{y}^2=\big[(\text{a}\sec\theta+\text{b}\tan\theta)^2+(\text{a}\tan\theta-\text{b}\sec\theta)^2\big]$
$=\big(\text{a}^2\sec^2\theta+\text{b}^2\tan^2\theta+2\text{ab}\sec\theta\tan\theta\big)\\ \ -\big(\text{a}^2\tan^2\theta+\text{b}^2\sec^2\theta+2\text{ab}\tan\theta\sec\theta\big)$
$=\text{a}^2\sec^2\theta+\text{b}^2\tan^2\theta-\text{a}^2\tan^2\theta+\text{b}^2\sec^2\theta$
$=\big(\text{a}^2\sec^2\theta+\text{a}^2\tan^2\theta\big)-\big(\text{b}^2\sec^2\theta+\text{b}^2\tan^2\theta\big)$
$=\text{a}^2\big(\sec^2\theta-\tan^2\theta\big)-\text{b}^2\big(\sec^2\theta-\tan^2\theta\big)$
$=\text{a}^2-\text{b}^2$ $\big[\because\ \sec^2+\tan^2=1\big]$
Hence, $\text{x}^2-\text{y}^2=\text{a}^2-\text{b}^2$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Trigonometric Identities→Trigonometric Identities — all question types→Maths STD 10 question papers→CBSE Board STD 10 question papers→CBSE Board question paper generator→

Similar questions

Show that the cube of a positive integer of the form 6q + r, q is an integer and r = 0, 1, 2, 3, 4, 5 is also of the form 6m + r.
→
Solve the following system of linear equation graphically and shade the region between the two lines and x-axis:
3x + 2y -11 = 0,
2x - 3y + 10 = 0.
→
If (2p + 1), 13, (5p - 3) are in AP, find the value of p.
→
Solve for x and y:
$\frac{5}{\text{x}+\text{1}}-\frac{2}{\text{x}-\text{1}}=\frac{1}{2}$
$\frac{10}{\text{x}+\text{1}}+\frac{2}{\text{y}-\text{1}}=\frac{5}{2}$
→
In a $\triangle\text{ABC},\angle\text{C}=90^\circ,\angle\text{ABC}=\theta^\circ,\text{BC}=21\text{units}$ and AB = 29units.
Show that $\big(\cos^2\theta-\sin^2\theta\big)=\frac{41}{841}.$
→
Solve the following quadratic equation:
$\Big(\frac{\text{x}}{\text{x}}+1\Big)^2-5\Big(\frac{\text{x}}{\text{x}+1}\Big)+6=0,$ $\text{x}\neq-1$
→
If the zeros of the polynomial $f(x) = 2x^3 - 15x^2 + 37x - 30$ are in A.P., find them.
→
A sailor goes 8km downstream in 40 minutes and returns in 1 hours. Determine the speed of the sailor in still water and the speed of the current.
→
The sum of the numerator and denominator of a fraction is 4 more than twice the numerator. If the numerator and denominator are increased by 3, they are in the ratio 2 : 3. Determine the fraction.
→
Prove that the points (2a, 4a), (2a, 6a) and $(2\text{a}+\sqrt{3}\text{a}, 5\text{a})$ are the vertices of an equilateral triangle.
→